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2019-10-14T11:48:38Z

Created page with "{{Dimensions |nd = 1 |nx = 2 |nw = 1 |nre = 2 |nri = 1 }}Th..."

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{{Dimensions
|nd = 1
|nx = 2
|nw = 1
|nre = 2
|nri = 1
}}This site describes a Lotka Volterra variant where a terminal inequality constraint on the differential states is added. A violation of this constraint is penalized as part of the Mayer objective.

== Mathematical formulation ==

The mixed-integer optimal control problem is given by

<p>
<math>
\begin{array}{llclr}
\displaystyle \min_{x, w, s} & x_2(t_f) + 10s \\[1.5ex]
\mbox{s.t.}
& \dot{x}_0 & = & x_0 - x_0 x_1 - \; c_0 x_0 \; w, \\
& \dot{x}_1 & = & - x_1 + x_0 x_1 - \; c_1 x_1 \; w, \\
& \dot{x}_2 & = & (x_0 - 1)^2 + (x_1 - 1)^2, \\[1.5ex]
& x_0 & \geq & 1.1 - s, \\
& x(0) &=& (0.5, 0.7, 0)^T, \\
& w(t) &\in& \{0, 1\}, \\
& s & \geq & 0.
\end{array}
</math>
</p>

Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. This problem variant penalizes a biomass x(0) that is below 1.1 at the end of the time horizon.

== Parameters ==

These fixed values are used within the model.

<math>
\begin{array}{rcl}
[t_0, t_f] &=& [0, 12],\\
(c_{0}, c_{1}) &=& (0.4, 0.2),
\end{array}
</math>

== Reference Solutions ==

If the problem is relaxed, i.e., we demand that <math>w(t)</math> is in the continuous interval <math>[0, 1]</math> rather than being binary, the optimal solution can be determined by means of direct optimal control.

The optimal objective value of the relaxed problem with <math> n_t=12000, \, n_u=200 </math> is <math>x_2(t_f) =1.36548113</math>. The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is <math>x_2(t_f) =1.38756111</math>.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:Lotka_term_ineq_Relaxed_12000_200.pdf| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=12000, \, n_u=200</math>.
Image:Lotka_term_ineq_CIA_12000_200.pdf| Differential states and binary cotnrol determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=200</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
</gallery>


[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Tracking objective]]
[[Category:Chattering]]
[[Category:Sensitivity-seeking arcs]]
[[Category:Population dynamics]]

Lotka Volterra (terminal constraint violation) - Revision history

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